| L(s) = 1 | + 12·5-s + 8·13-s + 62·25-s − 24·29-s − 48·41-s − 14·49-s − 16·61-s + 96·65-s − 24·73-s + 20·97-s − 60·101-s + 22·121-s + 168·125-s + 127-s + 131-s + 137-s + 139-s − 288·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 5.36·5-s + 2.21·13-s + 62/5·25-s − 4.45·29-s − 7.49·41-s − 2·49-s − 2.04·61-s + 11.9·65-s − 2.80·73-s + 2.03·97-s − 5.97·101-s + 2·121-s + 15.0·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 23.9·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.310609946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.310609946\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - 6 T + 23 T^{2} - 66 T^{3} + 156 T^{4} - 66 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 + 2 p T^{2} + 73 T^{4} + 50 p T^{6} + 3124 T^{8} + 50 p^{3} T^{10} + 73 p^{4} T^{12} + 2 p^{7} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 - 2 p T^{2} + 217 T^{4} - 50 p T^{6} - 1916 T^{8} - 50 p^{3} T^{10} + 217 p^{4} T^{12} - 2 p^{7} T^{14} + p^{8} T^{16} \) |
| 13 | \( ( 1 - 4 T + 10 T^{2} + 80 T^{3} - 341 T^{4} + 80 p T^{5} + 10 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 38 T^{2} + 915 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 12 T + 86 T^{2} + 456 T^{3} + 1995 T^{4} + 456 p T^{5} + 86 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 + 2 p T^{2} + 1561 T^{4} + 722 p T^{6} + 594484 T^{8} + 722 p^{3} T^{10} + 1561 p^{4} T^{12} + 2 p^{7} T^{14} + p^{8} T^{16} \) |
| 37 | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 24 T + 314 T^{2} + 2928 T^{3} + 21075 T^{4} + 2928 p T^{5} + 314 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 116 T^{2} + 6778 T^{4} + 345680 T^{6} + 16296739 T^{8} + 345680 p^{2} T^{10} + 6778 p^{4} T^{12} + 116 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 - 100 T^{2} + 4618 T^{4} - 96400 T^{6} + 1996243 T^{8} - 96400 p^{2} T^{10} + 4618 p^{4} T^{12} - 100 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 26 T^{2} + 387 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 148 T^{2} + 11002 T^{4} - 583120 T^{6} + 30488419 T^{8} - 583120 p^{2} T^{10} + 11002 p^{4} T^{12} - 148 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( 1 + 20 T^{2} + 922 T^{4} - 190000 T^{6} - 21674237 T^{8} - 190000 p^{2} T^{10} + 922 p^{4} T^{12} + 20 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 52 T^{2} + 7302 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 6 T + 131 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 154 T^{2} + 17475 T^{4} + 154 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 74 T^{2} - 7271 T^{4} - 76294 T^{6} + 77303524 T^{8} - 76294 p^{2} T^{10} - 7271 p^{4} T^{12} + 74 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 188 T^{2} + 21222 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 19 T + p T^{2} )^{4}( 1 + 14 T + p T^{2} )^{4} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.62838229266459428246985932833, −3.51383258288459783778970509195, −3.42108121120288695209066745873, −3.37018557592179588886116289584, −3.25399911498659486258832417764, −3.06528379128232454465172502649, −2.89199085155089760384074389969, −2.79054597002352348861306772571, −2.72729507804005166323765089301, −2.66735245335462995189307142728, −2.22853168912669476104606924208, −2.17526607836568666722737585808, −2.10283844809380936787483375560, −1.90366910308007887423415309903, −1.84543232434134857179203904316, −1.82572965913131229714887071100, −1.72988423699714022868361600197, −1.51618139158738222713680947617, −1.43247724401745467803188392050, −1.39460184917397311088589556024, −1.32613820572042207504646281302, −1.14252676869518977892311054143, −0.51885840453874058369124832448, −0.36743271443800158687692803306, −0.06602528640683071258317304624,
0.06602528640683071258317304624, 0.36743271443800158687692803306, 0.51885840453874058369124832448, 1.14252676869518977892311054143, 1.32613820572042207504646281302, 1.39460184917397311088589556024, 1.43247724401745467803188392050, 1.51618139158738222713680947617, 1.72988423699714022868361600197, 1.82572965913131229714887071100, 1.84543232434134857179203904316, 1.90366910308007887423415309903, 2.10283844809380936787483375560, 2.17526607836568666722737585808, 2.22853168912669476104606924208, 2.66735245335462995189307142728, 2.72729507804005166323765089301, 2.79054597002352348861306772571, 2.89199085155089760384074389969, 3.06528379128232454465172502649, 3.25399911498659486258832417764, 3.37018557592179588886116289584, 3.42108121120288695209066745873, 3.51383258288459783778970509195, 3.62838229266459428246985932833
Plot not available for L-functions of degree greater than 10.
